Topology

From Citizendium, the Citizens' Compendium
Jump to: navigation, search
This article is a stub and thus not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

Topology (from Greek τόπος - "place" and λόγος - "study") is a branch of mathematics that studies the propeties of objects that are preserved through continuous deformations (such as stretching, bending and compression). "Continuous" means that ruptures (breaking of the object's integrity) and splicing or glueing (matching object's points) do not happen. Topology grew out of geometry, but unlike most areas of geometry, topology is not concerned with metric properties such as distances between points. Instead, topology involves the study of properties that describe how a space is assembled, such as connectedness and orientability. Topology may be viewed as the search for solutions of problems relating to the geometry of position in the true sense of the term.

History

Topology started to develop in 18th century, when Leonhard Euler solved Seven Bridges of Königsberg in 1736. The term "topology" was firstly used by Johann Benedict Listing. Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. Henri Poincaré published Analysis Situs in 1895, introducing the concepts of homotopy and homology, which are now considered part of algebraic topology.

Properties

Two main properties of objects studied in topology are homeomorphism and homotopy equivalence.